While the words sense and meaning are considered synonyms (please see the entry for meaning from The American Heritage Dictionary of the English Language (Fourth Edition) -- for one source), the two words do have subtle distinctions and associated connotations. Generally speaking, the word sense, due to its derivation, is associated more with perceptions arising from the senses.
Thus, you're right to find that the word "meaning" does not work quite as well as "sense" in the example you have provided.
In any event, it's apparent that your sense of the meaning involved here is clear.
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